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Einstein Ring Hoodie

ONE GALAXY, SEEN AS A RING | Pixelated Physics

\[\beta = \theta - \frac{\theta_E^2}{\theta}\]
$54.99 USD
$54.99 USD
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Size guide S–5XL · inches

Mass bends light. Line a distant galaxy up behind a nearer one and the light arrives from several directions at once. A background galaxy lensed by a point mass into arcs and a ring, from the exact lens equation β = θ − θE²/θ. Mass is the lens. Pixel-art unisex hoodie in black; equation, symbols and sources in the physics panel below.

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Details

Size & fitS–5XL

Unisex Heavy Blend hoodie (Gildan 18500), classic fit.

SizeWidthLengthSleeve
S202733.5
M222834.5
L242935.5
XL263036.5
2XL283137.5
3XL303238.5
4XL323339.5
5XL343440.5

Measurements in inches. Width and length ±1 in, sleeve (from center back) ±0.75 in.

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Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

A mass lying exactly between us and a distant source bends the source's light into a ring. The Einstein radius of that ring depends on the lens mass and the distances involved, so measuring it weighs the lens.

Mass bends light. Line a distant galaxy up behind a nearer one and the light arrives from several directions at once.

For a point mass the mapping is exact and short: β = θ − θE²/θ. A source exactly behind the lens becomes a ring of radius θE. Move it slightly off axis and the ring breaks into two bright arcs, one on each side.

Lensing keeps surface brightness, so the art was made by tracing each pixel back through the lens equation to the source galaxy and copying its brightness. The amber dot is the lens.

Einstein worked out the ring in 1936 and thought it would never be seen. The first ring was found in radio data in 1988, and a complete infrared ring in 1998.

One galaxy, seen as a ring.

Equations

\[\beta = \theta - \frac{\theta_E^2}{\theta}\]
\[\theta_E^2 = \frac{4GM}{c^2}\,\frac{D_{ls}}{D_l\,D_s}\]
SymbolMeaningUnit
\(\beta\) true angular position of the source \(\mathrm{rad}\)
\(\theta\) apparent angular position of the image \(\mathrm{rad}\)
\(\theta_E\) Einstein radius \(\mathrm{rad}\)
\(G\) gravitational constant \(\mathrm{m^3·kg^{-1}·s^{-2}}\)
\(M\) lens mass \(\mathrm{kg}\)
\(c\) speed of light \(\mathrm{m/s}\)
\(D_l, D_s, D_{ls}\) distances to the lens, to the source, and from lens to source \(\mathrm{m}\)

Sources

  1. Einstein (1936) Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field, Science 84, 506
  2. Schneider, Ehlers & Falco, Gravitational Lenses (1992)
  3. King et al. (1998) A complete infrared Einstein ring in the gravitational lens system B1938+666, MNRAS 295, L41

Source Einstein, A. (1936). Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field. Science 84, 506–507.doi.org/10.1126/science.84.2188.506

Field notes

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